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Nancy Pfenning
nancyp+@pitt.edu
  

Statistics in a Modern World 800
Assignment 4

Homework Exercises Assigned from Part 4 (60 pts.) due Mon. Nov. 18 in Lecture

CHAPTER 18

#2 (3 pts.) Suppose the proportion of students who are left-handed is .12, and you take a random sample of 200 students. Use the Rule for Sample Proportions to draw a picture similar to Figure 18.3 p.322, showing the possible sample proportions for this situation.

 

 

 

 

 

#5 (3 pts.) Suppose you are interested in estimating the average number of miles per gallon of gasoline your car can get. You calculate the miles per gallon for each of the next 9 times you fill the tank. Suppose, in truth, the values for your car are bell-shaped, with a mean of 25 miles per gallon and a standard deviation of 1. Draw a picture similar to Figure 18.3 of the possible sample means you are likely to get based on your sample of 9 observations. (Include the intervals into which 68%, 95%, and almost all of the potential sample means will fall.)

 

 

 

 

 

 

#6. (3 pts.) Refer to Exercise 5. Redraw the picture under the assumption that you will collect 100 measurements instead of only 9.

 

 

 

 

 

#14 (2 pts.) Tell whether each of the following situations meets the conditions (pp.319-320) for which the Rule for Sample Proportions applies. If not, tell which condition is violated.

  1. You are interested in knowing what proportion of days in typical years have rain or snow in the area where you live. For the months of January and February, you record whether there is rain or snow each day, and then you calculate the proportion.

    2. (i) meets conditions (ii) violates condition #1 (iii) violates #2 (iv) violates #3

  2. A large company wants to determine what proportion of its employees are interested in on-site day care. The company asks a random sample of 100 employees and calculates the sample proportion who are interested.

    3. (i) meets conditions (ii) violates condition #1 (iii) violates #2 (iv) violates #3

 

#15 (3 pts.) Tell whether each of the following situations meets the conditions (p.323) for which the Rule for Sample Means applies.

  1. A researcher is interested in what the average cholesterol level would be if people restricted their fat intake to 30% of calories. He gets a group of patients who have had heart attacks to volunteer to participate, puts them on a restricted diet for a few months, and then measures their cholesterol. (i) applies (ii) does not apply
  2. A university wants to know the average income of its alumni. Staff members select a random sample of 200 alumni and mail them a questionnaire. They follow up with a phone call to those who do not respond within 30 days. (i) applies (ii) does not apply
  3. An automobile manufacturer wants to know the average price for which used cars of a particular model and year are selling in a certain state. They are able to obtain a list of buyers from the state motor vehicle division, from which they select a random sample of 20 buyers. They make every effort to find out what those people paid for the cars and are successful in doing so. (i) applies (ii) does not apply

CHAPTER 19

#1 (4 pts.) An advertisement for Seldane-D, a drug prescribed for seasonal allergic rhinitis, reported results of a double-blind study in which 374 patients took Seldane-D and 193 took a placebo (Time, 27 March 1995, p.18). Headaches were reported as a side effect by 65 of those taking Seldane-D.

  1. What is the sample proportion of Seldane-D users who reported headaches?
  2. What is the standard deviation (standard error) for the proportion computed in part (a)?
  3. Construct a 95% confidence interval for the population proportion based on the information from parts (a) and (b).

    4.  

     

  4. Circle the best interpretation of your confidence interval:

    5. (i) 95% of the time, in the long run, such a confidence interval will contain true population proportion

    (ii) 95% of sample proportions fall in this interval

    (iii) the probability is 95% that true population proportion will fall in this interval

    (iv) the probability is 95% that sample proportion will fall in this interval

#5. (2 pts.) What level of confidence would accompany each of the following intervals?

  1. Sample proportion plus or minus 1.0 standard errors
  2. Sample proportion plus or minus 1.645 standard errors
  3. Sample proportion plus or minus 1.96 standard errors
  4. Sample proportion plus or minus 2.576 standard errors

#6 (3 pts.) Tell whether the width of a confidence interval would increase, decrease, or remain the same as a result of each of the following changes:

  1. Sample size is doubled, from 400 to 800. (i) increase (ii) decrease (iii) same
  2. Population size doubled from 25 million to 50 million (i)increase (ii)decrease (iii)same
  3. Level of confidence is lowered from 95% to 90% (i) increase (ii) decrease (iii) same

#11 (4 pts.) A university is contemplating switching from the quarter system to the semester system. The administration conducts a survey of a random sample of 400 students and finds that 240 of them prefer to remain on the quarter system.

  1. Construct a 95% confidence interval for the true proportion of all students who would prefer to remain on the quarter system.

    2.  

  2. Does the interval you computed in part (a) provide convincing evidence that the majority of students prefer to remain on the quarter system? Answer yes or no.
  3. Now suppose that only 50 students had been surveyed that that 30 said they preferred the quarter system. Compute a 95% confidence interval for the true proportion who prefer to remain on the quarter system.

    4.  

  4. Does the interval you computed in part (c) provide convincing evidence that the majority of students prefer to remain on the quarter system? Answer yes or no.

CHAPTER 20

#1 (1 pt.) In Chapter 19, we saw that to construct a confidence interval for a population proportion it was enough to know the sample proportion and the sample size. In constructing a confidence interval for population mean, what else is needed besides sample mean and sample size?

#3 (2 pts.) The Baltimore Sun (Haney, 21 February 1995) reported on a study by Dr. Sara Harkness, in which she compared the sleep patterns of 6-month-old infants in the U.S. and the Netherlands. She found that the 36 U.S. infants slept an average of just under 13 hours a day, whereas the 66 Dutch infants slept an average of almost 15 hours.

  1. Suppose the standard deviation was 0.5 hour for each group. Compute the standard error of the mean (SEM) for the U.S. babies.
  2. Continuing to assume that the standard deviation is 0.5 hour, compute a 95% confidence interval for the mean sleep time for 6-month-old babies in the U.S.

 

#4 (1 pt.) What is the probability that a 95% confidence interval will not cover the true population value?

#10 (1 pt.) In Case Study 6.4 p.101, which examined maternal smoking and child’s IQ, one of the results reported in the journal article was the average number of days the infant spent in the neonatal intensive care unit. The results showed an average of 0.35 day for infants of nonsmokers and an average of 0.58 day for the infants of women who smoked ten or more cigarettes per day. In other words, the infants of smokers spent an average of 0.23 day more in neonatal intensive care. A 95% confidence interval for the difference in the two means extended from —3.02 days to +2.57 days. To explain why it would have been misleading to report, "These results show that the infants of smokers spend more time in neonatal intensive care than do the infants of nonsmokers" complete the following sentence: Such a report would be misleading because the confidence interval contains  …  (fill in the blank with a single word or number)

CHAPTER 21

#11 (2 pts.) Given the convention of declaring that a result is statistically significant if the p-value is 0.05 or less, what decision would be made in each case?

  1. p-value = 0.35 (i) reject the null hypothesis (ii) do not reject the null hypothesis
  2. p-value = 0.04 (i) reject the null hypothesis (ii) do not reject the null hypothesis

#12a (1 pt.) In previous chapters, we learned that researchers have discovered a link between vertex baldness and heart attacks in men. State the null and alternative hypotheses used to investigate whether there is such a relationship.

 

 

#14 (3 pts.) Suppose that a study is designed to choose between the hypotheses:

Null hypothesis: Population proportion is 0.25

Alternative hypothesis: Population proportion is higher than 0.25

On the basis of a sample of size 500, the sample proportion is 0.29. The standard deviation for the potential sample proportions in this case is about 0.02.

  1. Compute the z statistic corresponding to the sample proportion of 0.29, assuming the null hypothesis is true.

    2.  

  2. What is the p-value?
  3. Based on the results of parts (a) and (b), make a conclusion: (i) population proportion may be 0.25 (ii) population proportion is higher than 0.25

CHAPTER 22

#1 (3 pts.) In Exercise 12 in Chapter 19, we learned that in a survey of 507 adult American Catholics, 59% answered yes to the question, "Do you favor allowing women to be priests?"

  1. Set up the null and alternative hypotheses for deciding whether a majority of American Catholics favor allowing women to be priests.

    2.  

     

  2. Using Example 3 p.392 as a guide, compute the test statistic for this situation.

    3.  

     

     

  3. If you have done everything correctly, the p-value for the test is about 0.00005. Based on this, make a conclusion for this situation: (i) A majority of American Catholics favor allowing women to be priests (ii) A majority of American Catholics do not necessarily favor allowing women to be priests

#2 (1 pt. ) Refer to Exercise 1. Is the test described there a one-sided or a two-sided test?

#3 (1 pt.) Suppose a one-sided test for a proportion resulted in a p-value of 0.03. What would the p-value be if the test were two-sided instead?

#4 (2 pts.) Say a two-sided test for difference in two means resulted in a p-value of 0.08.

  1. Using the usual criterion for hypothesis testing, would we conclude that there was a difference in the population means? Answer yes or no.
  2. Suppose the test had been constructed as a one-sided test instead, and the evidence in the sample means was in the direction to support the alternative hypothesis. Using the usual criterion for hypothesis testing, would we be able to conclude that there was a difference in the population means? Answer yes or no.

#5 (1 pt.) Suppose you were given a hypothesized population mean, a sample mean, a sample standard deviation, and a sample size for a study involving a random sample from one population. What would you use as the test statistic? (i) z (ii) t (iii) chi-square

#6 (3 pts.) For each z statistic, use normal tables (p.137) to find the p-value for each of these examples taking into account whether the test is one-sided or two-sided:

  1. Chapter 22, Example 2, z=2.17, two-sided test
  2. Chapter 21, Example 1, z=2.00, one-sided test
  3. Case Study 21.1, z=4.09, one-sided test

#9 (2 pts.) On July 1, 1994, The Press of Atlantic City, NJ, had a headline reading, "Study: Female hormone makes mind keener" (p. A2) Here is part of the report:

"Halbreich said he tested 36 post-menopausal women before and after they started the estrogen therapy. He gave each one a battery of tests that measured such things as memory, hand-eye coordination, reflexes and the ability to learn new information and apply it to a problem. After estrogen therapy started, he said, there was a subtle but statistically significant increase in the mental scores of the patients."

  1. State the appropriate null and alternative hypotheses.

    2.  

  2. What type of study is this? (i) matched pairs (ii) two-sample

CHAPTER 23

#2a (1 pt.) Refer to Case Study 23.2 p.415, in which a report stated that Internet use was associated with a statistically significant increase in drpression. Which type of hypothesis test would have been more appropriate? (i) one-sided (ii) two-sided

#17 (1 pt.) Would it be easier to reject hypotheses about populations that had a lot of natural variability in the measurements or a little variability in the measurements?

(i) a lot (ii) a little

CHAPTER 22

#17 (7 pts.) On January 30, 1995, Time magazine reported the results of a poll of adult Americans, in which they were asked, "Have you ever driven a car when you probably had too much alcohol to drive safely?" The exact results were not given, but from the information provided we can guess at what they were. Of the 300 men who answered, 189 (63%) said yes and 108 (36%) said no. The remaining 3 weren’t sure. Of the 300 women, 87 (29%) said yes while 210 (70%) said no, and the remaining 3 weren’t sure.

(SHOW YOUR WORK FOR THE FOLLOWING BELOW ON THIS PAGE)

  1. Organize the data into a two-by-two table, ignoring the "not sure" answers.
  2. State the appropriate null and alternative hypotheses.
  3. Set up a table of numbers expected if the null hypothesis were true.
  4. Set up a table comparing the observed and expected numbers, as on pp. 202-203.
  5. Calculate the chi-squared statistic.
  6. The p-value is (i) more than .05 (ii) less than .05.
  7. State your conclusion: there (i) IS (ii) IS NOT NECESSARILY a relationship between gender and response to the question.

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